2
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This is Proposition 2.2, page 7, (Simplicial Homotopy Theory, Paul G. Goerss & John F. Jardine). I understand the reasons for the first and the third isomorphisms, but I could not figure out the reason for the second isomorphism. Any help is very much appreciated.

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That's the definition of $S(Y)$: its $n$-simplices are the maps $|\Delta^n|\to Y$. Note there's a Yoneda lemma in here too: $n$-simplices of a simplicial set are morphisms from $\Delta^n$.

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    ... and the trick of presheaves that $F \cong \operatorname{Nat}(\mathbf{y}(-), F)$. – 2017-01-23